## Verifying a vector identity using Levi-Civita

VERIFYING A VECTOR IDENTITY USING LEVI-CIVITA

**Bianca Rae B. Sambo**

Hello physics enthusiast! I am BR, a graduate student in Physics in Mindanao State University – Iligan Institute of Technology. Hope my solution will be of use to you. Keep visiting this site!

The vector identity to be verified using Levi-Civita is,

where we define the following vectors as,

We start by expanding the left hand side of the identity.

Then we set i = j and apply the distributive property of the partial derivative.

Equation 1:

Notice in the right hand side of the vector identity, we have the equations and . These two equations can be written as,

and

So we incorporate this to Equation 1 above.

Equation 2:

Notice also that:

Similarly,

So plugging these equations to equation 2, we finally verified the vector identity because we get: